Let $R = \mathbb{C}[x_1,…,x_n]$, let $J\subset R$ be a graded ideal, and consider the initial monomial ideal $\operatorname{in}(J)$ with respect to some term order. Suppose that we are given a linear ...

About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer.
Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full ...

For rings finitely-generated over a field, the theory of Groebner bases gives us quite an efficient algorithm for determining whether an element of the ring is in a given ideal of the ring.
Is there ...

Hello,
Could you help me with a reference to elementary properties of Groebner bases in rings of formal power series over a field? I am especially interested in generic initial ideals.
Thank you in ...

Is there an easy way to determine whether a set of elements in a field generates the whole field or only a subfield?
Specifically, I have a subfield of $k(x,y)$ described in terms of a canonical set ...

Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and
$C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated ...

Gröbner bases are usually considered in the ring of polynomials over a field. However, there are useful definitions and algorithms for Gröbner bases over other coefficient rings; see, for instance, ...

Let $A$ and $B$ be two polynomials in $\mathbf Z[X]$ which generate $\mathbf Z[X]$, that is assume that there exist polynomials $U$ and $V$ in $\mathbf Z[X]$ such that
$$
A \cdot U + B \cdot V=1.
$$
...

This question comes from the explicit construction of a smooth projective model of a hyperelliptic curve. Nevertheless it is fully elementary and, to me, more interesting than hyperelliptic curves.
...

Suppose we have a finite set of generators for an ideal $I \subset R := \Bbbk[x_1,\dotsc, x_n]$, where $\Bbbk$ is a field. If we choose a monomial ordering, then Buchberger's algorithm allows us to ...

Given a system of multivariate, polynomial equations, is there a way to determine if it has a solution in a given field (for instance the set of all reals). I don't care what the solution is, I just ...

Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-Lie algebra, which is not necessarily free as a $k$-module. Let $S\left(L\right)$ denote the symmetric algebra of $L$ (over $k$), constructed ...

Are there any articles/books/examples where a non-standard monomial order is used?
What are the applications of these monomial orders? In particular, uses in groebner bases and variable elimination.
...

Hi all,
I'm an engineer assigned to determine some parameters of a manipulator (ie., calibration). It has a number of parameters, but after some manipulations of its dynamic equations, I can have the ...